$A$ die is thrown. Find the probability of the following event: $A$ number more than $6$ will appear.

Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^{N} < N!$ is $\frac{m}{n}$,where $m$ and $n$ are coprime,then $4m - 3n$ is equal to $......$.

Boxes $B_1, B_2$,and $B_3$ contain balls as given below:

Box	White	Black
$B_1$	$1$	$2$
$B_2$	$3$	$1$
$B_3$	$2$	$3$

One ball is drawn at random from each box. Then,among the balls drawn,the probability that two are black and one is white,is

If three students $A, B, C$ independently solve a problem with probabilities $\frac{1}{3}, \frac{1}{4}$ and $\frac{1}{5}$ respectively,then the probability that the problem will be solved is

Two dice are rolled one after the other. The probability that the number on the first die is smaller than the number on the second die is

An experiment consists of tossing a coin and then throwing it a second time if a head occurs. If a tail occurs on the first toss,then a die is rolled once. Find the sample space.